Combinatorics Through Guided Discovery, 2004a

2.3. Graphs and Trees 47
2.3.6 The deletion/contraction recurrence for spanning trees
There are two operations on graphs that we can apply to get a recurrence (though
a more general kind than those we have studied for sequences) which will let us
compute the number of spanning trees of a graph. The operations each apply to an
edge e of a graph G. The first is called deletion;wedelete the edge e from the graph
by removing it from the edge set. Figure 2.3.4 shows how we can delete edges from
a graph to get a spanning tree.
Figure 2.3.4: Deleting two appropriate edges from this graph gives a spanning tree.
The second operation is called contraction.
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Figure 2.3.5: The results of contracting three different edges in a graph.
Contractions of three different edges in the same graph are shown in Figure 2.3.5.
Intuitively, we contract an edge by shrinking it in length until its endpoints coincide;
we let the rest of the graph “go along for the ride.” To be more precise, we contract
the edge e with endpoints v and w as follows:
1. remove all edges having either v or w or both as an endpoint from the edge
set,
2. remove v and w from the vertex set,
3. add a new vertex E to the vertex set,
4. add an edge from E to each remaining vertex that used to be an endpoint of
an edge whose other endpoint was v or w, and add an edge from E to E for
any edge other than e whose endpoints were in the set {v, w}.
We use G − e (read as G minus e) to stand for the result of deleting e from G,
and we use G/e (read as G contract e) to stand for the result of contracting e from
G.